Clarence Irving Lewis (1883-1964) was an American logician who, in 1918, published an influential book about logic, A Survey of Symbolic Logic. I was reading it as part of the Journal of the History of Philosophy‘s Master Class on Frege with Juliet Floyd. I was struck by Lewis how distinguishes two views of logic, the “orthodox” and “heterodox” (pg 354):
The differences between the “orthodox” and this “heterodox” view have to do principally with two questions: (1) What is the nature of the fundamental operations in mathematics; are they essentially of the nature of logical inference and the like, or are they fundamentally arbitrary and extra-logical? (2) Is logistic ideally to be stated so that all its assertions are metaphysically true, or is its principal business the exhibition of logical types of order without reference to any interpretation or application? The two questions are related.
A fascinating thing about this passage is how Lewis sees the issue of whether theses of symbolic logic are synthetic a priori truths under an intended interpretation (as they would need to be to count as “metaphysically true”) as bound up with the issue of whether logicism is true. Putting aside my casting of the distinction Lewis is drawing, Lewis is claiming that there is an inverse relationship between the substantial truth of symbolic logic’s principles and mathematics lying outside logic: logic lacks substantially true principles if and only if mathematics lies outside logic.
That is striking to me because it is consonant with my view that logic being synthetic a priori and a genuine science is crucial for Russell. In 1914, Russell wants to make logic the essence of philosophy. He can hardly do that if logic lacks informative and non-tautologous principles such as would count as “metaphysically true” in Lewis’ sense (that is, as synthetic a priori in my casting of Lewis’ point).
There is much more to say, especially about what Lewis proceeds to say about taking logic as a symbolic apparatus amenable to various interpretations, and how doing so purges logic and mathematics of all meaning and reference. (Frege would be furious.) Lewis takes the upshot of this to be that one removes what is going on in the mind of a mathematician from mathematics. That is a point for another day. What I want to say is: this is all just to show that it is always philosophically fruitful to return to the classics.